Skip to main content
SpringerLink
Log in

Fractional Combinatorial Optimization

  • Reference work entry
  • First Online:
Handbook of Combinatorial Optimization

Abstract

This chapter considers combinatorial optimization problems with objective functions in the form of ratios of two functions. A parametric approach to such problems is described and two main general algorithmic methods—the Newton method and Megiddo’s parametric search—are explained, using in both cases the same example of computing maximum-ratio paths in acyclic graphs. Some general properties of these methods are presented, including a proof of a strongly polynomial bound on the number of iterations of the Newton method and a proof of the speedup of Megiddo’s parametric search obtained by using parallel algorithms. The power of these methods is further shown in detail analyses of an algorithm for the maximum mean-weight cut problem based on the Newton method and an algorithm for the maximum profit-to-time ratio cycle problem based on Megiddo’s parametric search.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 3,400.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Recommended Reading

  1. R.K. Ahuja, Minimum cost-reliability ratio path problem. Comput. Oper. Res. 15, 83–89 (1988)

    Article  MATH  Google Scholar 

  2. R.E. Bellman, On a routing problem. Q. Appl. Math. 16, 87–90 (1958)

    MATH  Google Scholar 

  3. A. Billionnet, Approximation algorithms for fractional knapsack problems. Oper. Res. Lett. 30(5), 336–342 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Blum, R.W. Floyd, V. Pratt, R.L. Rivest, R.E. Tarjan, Time bounds for selection. J. Comp. Syst. Sci. 7(4), 448–461 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Carlson, D. Eppstein, The weighted maximum-mean subtree and other bicriterion subtree problems. CoRR. (2005). abs/cs/0503023

    Google Scholar 

  6. P.J. Carstensen, The complexity of some problems in parametric, linear, and combinatorial programming. Ph.D. Thesis (Doctoral Thesis), Department of Mathematics, University of Michigan, Ann Arbor, 1983

    Google Scholar 

  7. R. Chandrasekaran, Minimum ratio spanning trees. Networks 7, 335–342 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Chandrasekaran, Y.P. Aneja, K.P.K. Nair, Minimal cost-reliability ratio spanning tree. Ann. Discrete Math. 11, 53–60 (1981)

    MathSciNet  MATH  Google Scholar 

  9. E. Cohen, N. Megiddo, Maximizing concave functions in fixed dimension, in Complexity in Numerical Optimization, ed. by P. Pardalos (World Scientific, Singapore, 1993), pp. 74–87

    Chapter  Google Scholar 

  10. E. Cohen, N. Megiddo, Strongly polynomial time and NC algorithms for detecting cycles in dynamic graphs. J. Assoc. Comput. Mach. 40(4), 791–830 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Cole, Parallel merge sort. SIAM J. Comput. 17(4), 770–785 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algorithms (MIT, Cambridge, 1990)

    MATH  Google Scholar 

  13. J.R. Correa, C.G. Fernandes, Y. Wakabayashi, Approximating a class of combinatorial problems with rational objective function. Math. Program. 124(1–2), 255–269 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. G.B. Dantzig, W.O. Blattner, M.R. Rao, Finding a cycle in a graph with minimum cost to time ratio with application to a ship routing problem, in Theory of Graphs, ed. by P. Rosentiehl (Gordon and Breach, New York, 1967), pp. 77–84

    Google Scholar 

  15. A. Dasdan, Experimental analysis of the fastest optimum cycle ratio and mean algorithms. ACM Trans. Des. Autom. Electron. Syst. 9, 385–418 (2004)

    Article  Google Scholar 

  16. A. Dasdan, R.K. Gupta, Faster maximum and minimum mean cycle algorithms for system-performance analysis. IEEE Trans. CAD Integr. Circ. Syst. 17(10), 889–899 (1998)

    Article  Google Scholar 

  17. A. Dasdan, S.S. Irani, R.K. Gupta, Efficient algorithms for optimum cycle mean and optimum cost to time ratio problems, in Proceedings of the 36th Annual ACM/IEEE Design Automation Conference(DAC ’99), ed. by M.J. Irwin (ACM, New York, 1999), pp. 37–42

    Google Scholar 

  18. W. Dinkelbach, On nonlinear fractional programming. Manag. Sci. 13, 492–498 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  19. T.R. Ervolina, S.T. McCormick, Two strongly polynomial cut cancelling algorithms for minimum cost network flow. Discrete Appl. Math. 46, 133–165 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. L.R. Ford Jr., D.R. Fulkerson, Flows in Networks. (Princeton University Press, Princeton, 1962)

    MATH  Google Scholar 

  21. B. Fox, Finding minimal cost-time ratio circuits. Oper. Res. 17, 546–551 (1969)

    Article  Google Scholar 

  22. M.L. Fredman, R.E. Tarjan, Fibonacci heaps and their uses in improved network optimization algorithms. J. Assoc. Comput. Mach. 34, 596–615 (1987)

    Article  MathSciNet  Google Scholar 

  23. A.V. Goldberg, Scaling algorithms for the shortest paths problem. SIAM J. Comput. 24(3), 494–504 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. A.V. Goldberg, R.E. Tarjan, A new approach to the maximum flow problem. J. Assoc. Comput. Mach. 35, 921–940 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. A.V. Goldberg, R.E. Tarjan, Finding minimum-cost circulations by canceling negative cycles. J. Assoc. Comput. Mach. 36, 388–397 (1989)

    MathSciNet  Google Scholar 

  26. D. Gusfield, Sensitivity analysis for combinatorial optimization. Technical Report UCB/ERL M90/22, Electronics Research Laboratory, University of California, Berkeley, May 1980

    Google Scholar 

  27. D. Gusfield, Parametric combinatorial computing and a problem of program module distribution. J. Assoc. Comput. Mach. 30(3), 551–563 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Hartmann, J.B. Orlin, Finding minimum cost to time ratio cycles with small integral transit times. Networks 23, 567–574 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. S. Hashizume, M. Fukushima, N. Katoh, T. Ibaraki, Approximation algorithms for combinatorial fractional programming problems. Math. Program. 37, 255–267 (1987)

    Article  MathSciNet  Google Scholar 

  30. T. Ibaraki, Parametric approaches to fractional programs. Math. Program. 26, 345–362 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  31. H. Ishii, T. Ibaraki, H. Mine, Fractional knapsack problems. Math. Program. 13, 255–271 (1976)

    Article  MathSciNet  Google Scholar 

  32. K. Iwano, S. Misono, S. Tezuka, S. Fujishige, A new scaling algorithm for the maximum mean cut problem. Algorithmica 11(3), 243–255 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  33. S.N. Kabadi, Y.P. Aneja, An efficient, strongly polynomial, ε-approximation parametric optimization scheme. Inform. Process. Lett. 64(4), 173–177 (1997)

    Article  MathSciNet  Google Scholar 

  34. R.M. Karp, A characterization of the minimum cycle mean in a digraph. Discrete Math. 23, 309–311 (1978)

    MathSciNet  MATH  Google Scholar 

  35. N. Katoh, A fully polynomial time approximation scheme for minimum cost-reliability ratio problems. Discrete Appl. Math. 35(2), 143–155 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  36. V. King, S. Rao, R. Tarjan, A faster deterministic maximum flow algorithm. J. Algorithms 17(3), 447–474 (1994)

    Article  MathSciNet  Google Scholar 

  37. G.W. Klau, I. Ljubic, P. Mutzel, U. Pferschy, R. Weiskircher, The fractional prize-collecting Steiner tree problem on trees: extended abstract, in Algorithms – ESA 2003, Proceedings of the 11th Annual European Symposium, Budapest, 16–19 September 2003. Lecture Notes in Computer Science, vol. 2832 (Springer-Verlag, Berlin, Heidelberg, 2003), pp. 691–702

    Google Scholar 

  38. E.L. Lawler, Optimal cycles in doubly weighted directed linear graphs, in Theory of Graphs, ed. by P. Rosentiehl (Gordon and Breach, New York, 1967), pp. 209–213

    Google Scholar 

  39. E.L. Lawler, Combinatorial Optimization: Networks and Matroids (Holt, Reinhart, and Winston, New York, 1976)

    MATH  Google Scholar 

  40. S.T. McCormick, A note on approximate binary search algorithms for mean cuts and cycles. UBC Faculty of Commerce Working Paper 92-MSC-021, University of British Columbia, Vancouver, 1992

    Google Scholar 

  41. N. Megiddo, Combinatorial optimization with rational objective functions. Math. Oper. Res. 4, 414–424 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  42. N. Megiddo, Applying parallel computation algorithms in the design of serial algorithms. J. Assoc. Comput. Mach. 30, 852–865 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  43. S. Mittal, A.S. Schulz, A general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into one, in APPROX-RANDOM. Lecture Notes in Computer Science, vol. 5171 (Springer-Verlag, Berlin, Heidelberg, 2008), pp. 179–192

    Google Scholar 

  44. C. Haibt Norton, S.A. Plotkin, É. Tardos, Using separation algorithms in fixed dimension. J. Algorithm 13, 79–98 (1992)

    Article  MATH  Google Scholar 

  45. J.B. Orlin, R.K. Ahuja, New scaling algorithms for the assignment and minimum mean cycle problems. Math. Program. 54, 41–56 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  46. C.H. Papadimitriou, K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity (Prentice-Hall, Englewood Cliffs, 1982)

    MATH  Google Scholar 

  47. P.M. Pardalos, A.T. Phillips, Global optimization of fractional programs. J. Global Opt. 1, 173–182 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  48. T. Radzik, Newton’s method for fractional combinatorial optimization, in Proceedings of the 33rd IEEE Annual Symposium on Foundations of Computer Science, Pittsburgh, PA, USA (1992), pp. 659–669

    Google Scholar 

  49. T. Radzik, Newton’s method for fractional combinatorial optimization. Technical Report STAN-CS-92-1406, Department of Computer Science, Stanford University, January 1992

    Google Scholar 

  50. T. Radzik, Parametric flows, weighted means of cuts, and fractional combinatorial optimization, in Complexity in Numerical Optimization, ed. by P. Pardalos (World Scientific, Singapore, 1993), pp. 351–386

    Chapter  Google Scholar 

  51. T. Radzik, A. Goldberg, Tight bounds on the number of minimum-mean cycle cancellations and related results. Algorithmica 11, 226–242 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  52. P. Rusmevichientong, Z.-J.M. Shen, D.B. Shmoys, A PTAS for capacitated sum-of-ratios optimization. Oper. Res. Lett. 37(4), 230–238 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  53. P. Rusmevichientong, Z.-J.M. Shen, D.B. Shmoys, Dynamic assortment optimization with a multinomial logit choice model and capacity constraint. Oper. Res. 58(6), 1666–1680 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  54. S. Schaible, Fractional programming 2. On Dinkelbach’s algorithm. Manag. Sci. 22, 868–873 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  55. S. Schaible, A survey of fractional programming, in Generalized Concavity in Optimization and Economics, ed. by S. Schaible, W.T. Ziemba (Academic, New York, 1981), pp. 417–440

    Google Scholar 

  56. S. Schaible, Bibliography in fractional programming. Z. Oper. Res. 26(7), 211–241 (1982)

    MathSciNet  MATH  Google Scholar 

  57. S. Schaible, Fractional programming. Z. Oper. Res. 27, 39–54 (1983)

    MathSciNet  MATH  Google Scholar 

  58. S. Schaible, T. Ibaraki, Fractional programming. Eur. J. Oper. Res., 12(4), 325–338 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  59. S. Schaible, J. Shi, Fractional programming: the sum-of-ratios case. Optim. Methods Softw. 18, 219–229 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  60. S. Schaible, J. Shi, Recent developments in fractional programming: single-ratio and max-min case, in Proceedings of the 3rd International Conference on Nonlinear Analysis and Convex Analysis, Tokyo, 25–29 August 2003 (Yokohama Publishers, Yokohama, 2004), pp. 493–506

    Google Scholar 

  61. P. Serafini, B. Simeone, Parametric maximum flow methods for minimax approximation of target quotas in biproportional apportionment. Networks 59(2), 191–208 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  62. M. Shigeno, Y. Saruwatari, T. Matsui, An algorithm for fractional assignment problems. Discrete Appl. Math. 56(2–3), 333–343 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  63. Y. Shiloach, U. Vishkin, An O(n 2logn) parallel max-flow algorithm. J. Algorithms 3, 128–146 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  64. S. Toledo, Maximizing non-linear concave functions in fixed dimension, in Complexity in Numerical Optimization, ed. by P. Pardalos (World Scientific, Singapore, 1993), pp. 429–447

    Chapter  Google Scholar 

  65. L.G. Valiant, Parallelism in comparison problems. SIAM J. Comput. 4, 348–355 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  66. C. Wallacher, A generalization of the minimum-mean cycle selection rule in cycle canceling algorithms. Unpublished manuscript, Institut für Angewandte Mathematik, Technische Universität Carolo-Wilhelmina, Germany, November 1989

    Google Scholar 

  67. Q. Wang, X. Yang, J. Zhang, A class of inverse dominant problems under weighted l ∞ norm and an improved complexity bound for Radzik’s algorithm. J. Global Optim. 34, 551–567 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  68. A.C. Yao, An O( | E | loglog | V | ) algorithm for finding minimum spanning trees. Inform. Process. Lett. 4, 21–23 (1975)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Informatics, Kings College London, London, UK

    Tomasz Radzik

Authors
  1. Tomasz Radzik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomasz Radzik .

Editor information

Editors and Affiliations

  1. Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Computer Science, University of Texas, Dallas, Richardson, Texas, USA

    Ding-Zhu Du

  3. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this entry

Cite this entry

Radzik, T. (2013). Fractional Combinatorial Optimization. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_62

Download citation

Publish with us

Policies and ethics

Search

Navigation